Self-conjugacy of abstract differential operators of the hyperbolic type

Sufficient conditions are obtained for the self-conjugacy of certain operators generated on a semiaxis or a complete axis by a differential expression of the form $l[y]=y''+ay-q(t)y$, where $A$ is a self-conjugate operator bounded below in a separable Hilbert space $H$, and, for almost all $t$, $q(t)$ is a bounded self-conjugate operator in $H$, locally summable with the square of the norm.